1️⃣ Relations and Functions
Relation: A relation from set A to set B is a subset of A \times B.
Function: A relation where each element of A has exactly one image in B.
Types of Functions
One–One (Injective)
Onto (Surjective)
Bijective (One–One + Onto)
Inverse Function
If f is bijective, then inverse exists:
f^{-1}(f(x)) = x
Composition of Functions
(f \circ g)(x) = f(g(x))
2️⃣ Inverse Trigonometric Functions
Example:
\sin^{-1}x,\ \cos^{-1}x,\ \tan^{-1}x
Important Identities
\sin^{-1}x + \cos^{-1}x = \frac{\pi}{2}
, \tan^{-1}x + \tan^{-1}\frac{1}{x} = \begin{cases} \frac{\pi}{2}, & x>0 \\ -\frac{\pi}{2}, &
x<0 \end{cases}
3️⃣ Matrices
A matrix is a rectangular array of numbers.
Example:
A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}
Types
Row matrix
Column matrix
Square matrix
Zero matrix
Identity matrix
Matrix Multiplication
(AB)_{ij} = \sum A_{ik}B_{kj}
4️⃣ Determinants
For a 2\times2 matrix:
Relation: A relation from set A to set B is a subset of A \times B.
Function: A relation where each element of A has exactly one image in B.
Types of Functions
One–One (Injective)
Onto (Surjective)
Bijective (One–One + Onto)
Inverse Function
If f is bijective, then inverse exists:
f^{-1}(f(x)) = x
Composition of Functions
(f \circ g)(x) = f(g(x))
2️⃣ Inverse Trigonometric Functions
Example:
\sin^{-1}x,\ \cos^{-1}x,\ \tan^{-1}x
Important Identities
\sin^{-1}x + \cos^{-1}x = \frac{\pi}{2}
, \tan^{-1}x + \tan^{-1}\frac{1}{x} = \begin{cases} \frac{\pi}{2}, & x>0 \\ -\frac{\pi}{2}, &
x<0 \end{cases}
3️⃣ Matrices
A matrix is a rectangular array of numbers.
Example:
A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}
Types
Row matrix
Column matrix
Square matrix
Zero matrix
Identity matrix
Matrix Multiplication
(AB)_{ij} = \sum A_{ik}B_{kj}
4️⃣ Determinants
For a 2\times2 matrix:
